CHAPITRE 5
Ecriture Matricielle du Modle Classique, Extension au Cas de K Rgresseurs
5.1 Introduction
Soit
Yt = 1 +2X2t + 3X3t + + KXKt + ut
Y1 = 1 +2X21 + 3X31 + + KXK1 + u1
..
YT = 1 +2X2T + 3X3T + + KXKT + uT
1 21 31 1 1 1
22 22 32 2 2
2 3 ( 1)( 1) ( ) ( 1)
1 ...
1 ......... ... ... ... ... ... ...
1 ...
K
K
TT T T KT K TT T K
Y X X X uuY X X X
uY X X X
K
= +
donc le modle scrit comme
(5.1) Y X= + u
5.2 Les Hypothses
1. E(u) = 0 !
1
2
( )
( ) 0
0( )( ) 0 ......
0( )T
Y X u E Y X
E u
E uE u
E u
= + =
= =
2. E(uuT) = 2I
22
1
2
1 2
21 1 2 1
22 1 2 2
21 2
21 1 2 1
0( ) ( )
( ) ......
....
...( )
... ... ... ...
...
( ) ( ) .... ( )
( )
Ti j
TT
T
T
TT
T T T
T
T
i jE uu I E u u
i j
uu
E uu E u u u
u
u u u u u
u u u u uE uu E
u u u u u
E u E u u E u u
EE uu
= = = = =
=2
2 1 2 2
21 2
2
22
2
( ) ( ) ... (... ... ... ...
( ) ( ) ... ( )
0 ... 0
0 ... 0( )
... ... ... ...
0 0 ...
T
T T
T
u u E u E u u
E u u E u u E u
E uu I
= =
)
T
3. (X) = K ! Les X sont LI ! (XTX)-1 existe 4. X est non-stochastique ! E(XTu) = 0 5. u N(0, 2I)
5.3 Estimation
(5.2)
Y X= +e
1
2 21 2
1
... ...
( ) ( )
( ) (
2 ( ) (
2( ) 2( ) 0
( )
( ) ( ) (
TT t
T
T T
T T T T T
T T T T T
TT T
T T
T T
ee
e e e e e e
e
e e Y X Y X
e e Y Y X Y Y X
e e Y Y X Y X X
e e X Y X X
X Y X X Equations Normal
X X X Y
= =
=
= +
= +
= + ==
=
1 ) ( )T TX X X X
) ( )
)
T T
T
X X
es
)
=
do
(5.3) 1 ( ) (OLS T TX X X Y =
Notez que XTe = 0. En effet
1
1
221 22 2 2
1 2
( ) ( )
( )( ) 0
:
1 1 ... 1 0
0....... ...... ... ... ... ...
...
T T T T
T T T T T
t
T t tT
TK K KT t kt
X e X Y X X Y X X
X e X Y X X X X X Y
Signification
eeeX X X e X
X e
eX X X e X
= = =
= =
= =
0
5.4 Thorme de GAUSS-MARKOV
5.4.1 Unbaisure
1 1
1 1
1
1
( ) ( ) (
( ) ( )
( )
( ) ( ) ( )
( )
T T T T
T T T T
T T
T T
X X X Y X X X X u
X X X X X X X u
X X X u
E X X X E u
E
= =
= +
= +
= + =
=
)+
5.4.2 Minimum Variance
1
1
1 1
2 21 1 2 2
1 1
1 1
( )
( )
( ) [ ][ ]
...
...
[( ) ][( ) ]
[( ) ( ) ]
(
T T
T T
T
K K
K K
T T T T T
T T T T
X X X u
X X X u
Par definition Var Cov E
E
E X X X u X X X u
E X X X uu X X X
X
= +
=
= =
=
=
=
= 1 1
2 1
2 1
1
1
) ( ) ( )
( )
( , ( ) )
:
( )
:
[( ) ]
0.
[ ]
(
T T T T
T
T
T T
T T
X X E uu X X X
X X
N X X
Soit un autre estimateur de
X X X Y AY
Considerons HY comme estimateur alternatif avec
HY X X X CY
ou C est une matrice
HY H X u HX Hu
E
=
= =
=
= = +
>
= = + = +
!
!
!
!
( )
1
1 1
2
)
[( ) ]
0
[ ][ ] [ ]
( ) ( )
( )
T T
T T T
T T T T T
T
ssi HX I
Mais
HX X X X C X I CX
HX I ssi CX
E E Huu H
E X X X C uu X X X C
Mais E uu I
= =
= + = +
= =
= =
= + + =
!
!
! !
2 1 1
2 1 1 1
2 1 2 1 2
( ) ( )
( ) ( ) ( )
( ) ( )
0
T T T T
T T T T T
T T T T
T
X X X C X X X C
X X CX X X X X X C CC
X X CC X X CC
Mais CC
= + + = + + + = + = + > >
!
!
!
!
T
5.5 Un Estimateur pour 2
Thorme Dans le modle linaire classique
Y = +t 1 2X2t + 3X3t + + KXKt + ut
2Te e
T K =
est un estimateur unbiais de 2.
En effet
( )
1 1
1
1
2
( ) ( )
( )
( )
( ), ( ... ) dim
( )
T T T T
T T
T T
T j
T T T T T T T
T
e Y X Y X X X X Y I X X X X Y MY
e I X X X X X u
e X X X X X X Mu
e Mu
M est symmetrique M M idempotente M M M de ension T T
e e u M Mu E e e E u M Mu E u Mu
Mais u Mu est u
.
= = = = = +
= +
=
= = =
= = =
" " "
( )
(1 )( )( 1)
2 21 1 1 2 3 2 1 2 2 4
2 21 2
1
1
.
(1 1)
(
( ) ( )
( ) ( ) ( )
( ) ( )
( ) ( ) ( )( )
(
T
T T T T
T T
T
T T
T T
T
n scalaire Eneffet
u M u
E u Mu E Tr u Mu E u X u u X u u X u X
E u Mu X X Tr M
Mais
Tr M Tr I Tr X X X X
Tr AB Tr BA
Tr M Tr I Tr X X X X T K
E e e
=
= = + + + = + =
=
=
= =
2
2
) ( ) ( )
( )
T
T
E u Mu T K
e eET K
= =
=
donc
Te e
T K (5.4)
est un estimateur sans-bais de . 2
5.1 IntroductionSoitYt = \(1 +\(2X2t + \(3X3t + + \(KXKt + utY1 = \(1 +\(2X21 + \(3X31 + + \(KXK1 + u1..YT = \(1 +\(2X2T + \(3X3T + + \(KXKT + uTdonc le modle scrit comme5.2 Les Hypothses1. E(u) = 0 (2. E(uuT) = (2INotez que XTe = 0. En effetThorme Dans le modle linaire classiqueYt = \(1 +\(2X2t + \(3X3t + + \(KXKt + ut
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